Don’t-care condition: meaning and examples

A don't-care condition is an input combination that never occurs, or whose output nobody needs, so the designer may treat its cell as 1 or 0, whichever gives a larger group. On the engine's map, don't-cares come only from stated assumptions.

In a truth table a don't-care is marked X or d, and in minterm notation it is listed separately, as in Σm(…) + d(…). The classic source is an encoding with unused codes: four bits can hold sixteen values, but a decimal digit uses only ten, so six combinations never arrive. Treating those cells as 1 when that enlarges a group, and as 0 otherwise, yields a shorter expression that agrees with the specification on every combination that can actually occur, and nothing is promised about the rest.

The artefact on this page is the BCD rule Σm(D, C, B, A; 6, 7, 8, 9) under the assumption NOT (D AND (C OR B)), which says codes 10 to 15 never occur. The map marks 6 cells as don't-care and reports 2 prime implicants; its cover is D OR (C AND B), with the group D taking cells 8 to 15 and the group C AND B taking cells 6, 7, 14 and 15. The result is labelled equivalent within the assumptions, and the scope annotation names assumption k1 as the source of the 6 don't-care cells.

Don't-cares appear wherever a domain has impossible states: unused opcodes, mutually exclusive flags, a banned user who cannot also be a member. They are also where errors hide, because an expression derived with them is wrong on the excluded cells if the exclusion was wrong. The term is confused with an unknown value and with an input that is simply absent from a formula; a don't-care is a deliberate, stated freedom.

Example

greater_than_five := Σm(D, C, B, A; 6, 7, 8, 9) · output

Assumptions: NOT (D AND (C OR B)) (The four bits encode a decimal digit 0 to 9 (BCD); codes 10 to 15 never occur.)

Exact checks

Always true (tautology)? No Counterexample: D = F, C = F, B = F, A = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: D = F, C = T, B = T, A = F · Exhaustively checked and solver cross-checked · 7 rows

Can be true? Yes Example assignment: D = F, C = T, B = T, A = F · Exhaustively checked and solver cross-checked · 7 rows

Are the assumptions consistent? Yes Example assignment: D = F, C = F, B = F, A = F · Exhaustively checked and solver cross-checked · 1 rows

Truth table (16 rows)

Don’t-care condition: meaning and examples
#	D	C	B	A	greater_than_five	k1	Scope
0	FALSE	FALSE	FALSE	FALSE	FALSE	TRUE	in scope
1	FALSE	FALSE	FALSE	TRUE	FALSE	TRUE	in scope
2	FALSE	FALSE	TRUE	FALSE	FALSE	TRUE	in scope
3	FALSE	FALSE	TRUE	TRUE	FALSE	TRUE	in scope
4	FALSE	TRUE	FALSE	FALSE	FALSE	TRUE	in scope
5	FALSE	TRUE	FALSE	TRUE	FALSE	TRUE	in scope
6	FALSE	TRUE	TRUE	FALSE	TRUE	TRUE	in scope
7	FALSE	TRUE	TRUE	TRUE	TRUE	TRUE	in scope
8	TRUE	FALSE	FALSE	FALSE	TRUE	TRUE	in scope
9	TRUE	FALSE	FALSE	TRUE	TRUE	TRUE	in scope
10	TRUE	FALSE	TRUE	FALSE	FALSE	FALSE	excluded
11	TRUE	FALSE	TRUE	TRUE	FALSE	FALSE	excluded
12	TRUE	TRUE	FALSE	FALSE	FALSE	FALSE	excluded
13	TRUE	TRUE	FALSE	TRUE	FALSE	FALSE	excluded
14	TRUE	TRUE	TRUE	FALSE	FALSE	FALSE	excluded
15	TRUE	TRUE	TRUE	TRUE	FALSE	FALSE	excluded

Plain-language reading

Beginnergreater_than_five is true exactly when the truth-table row of (D, C, B, A) is one of the rows 6, 7, 8, and 9.
In every other case, greater_than_five is false.
About the operators:
- Σm lists the truth-table rows (minterms) where the formula is true. Rows are numbered from 0 with the first listed input as the most significant bit.

Intermediategreater_than_five is true exactly when the truth-table row of (D, C, B, A) is one of the rows 6, 7, 8, and 9.
In every other case, greater_than_five is false.

Advancedgreater_than_five is true exactly when the truth-table row of (D, C, B, A) is one of the rows 6, 7, 8, and 9 [canonical sum of minterms (DNF)].
In every other case, greater_than_five is false.
Formal notation: Σm(D, C, B, A; 6, 7, 8, 9)

Karnaugh map

DC \ BA

0m0

0m1

0m3

0m2

0m4

0m5

1m7

1m6

Xm12

Xm13

Xm15

Xm14

1m8

1m9

Xm11

Xm10

cells m8, m9, m10, m11, m12, m13, m14, m15D
cells m6, m7, m14, m15C AND B

D OR (C AND B) · Derived D OR (C AND B). The groups use don't-care cells, so the result is only claimed within the assumptions. Equivalent on every assignment the assumptions allow (scoped: it was derived using the assumptions); no claim is made outside them.

6 cells excluded by the confirmed assumptions (k1) are marked don't-care. An expression derived with them is equivalent only within the assumptions.

The common slip

Treating a don't-care as a promise that the output is 0 there. D OR (C AND B) is true on cells 10 to 15, which the specification never mentions; it is correct only because those codes are assumed never to occur. Feed it code 12 and it answers TRUE.

Questions

Why does the engine refuse to guess don't-cares?

Because a don't-care is a claim about the domain, not about the function, and only the author can make it. The map marks a cell don't-care only when a stated assumption proves it unreachable, and then labels the result as equivalent within those assumptions.

Guides

Practice

Glossary

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.